Traditional methods in nanoparticle self-assembly utilize spherical cores as the anchoring motif. While successful in enabling directed control of interactions, spherical cores tend to restrict the range of accessible morphologies. This limitation serves as a significant hurdle for the establishment of self-assembly as a robust technique for both the design and fabrication of nanomaterials. Recently, experiments have began to utilize cores of varying shapes in order to expand on the current library of self-assembled morphologies. Using our theoretical models, we show that there exists a significant level of preferential partitioning to positions of varying curvature on the surface of anisotropic cores. This non-trivial graft distribution can be translated into different packing modalities, where the type of packing is not determined by the shape of the solid particle core, but rather by the emergent shape of the soft shell. These results highlight anisotropy as a powerful handle not only for accessing new morphologies, but also for a selective transition between different orientational packing for cores of the same shape type.